Please Help---Quantative Problems

[rehandossani]

1) For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n inclusive. If p is the smallest prime factor of h(100) + 1, then p is between
a. 2 and 10
b. 10 and 20
c. 20 and 30
d. 30 and 40
e. > 40


2) Positive integers x, y, and z are such that x is a factor of y and y is a factor of z, if z even? (EXTRA DIFFICULT)
a. xz is even
b. y is even

[ashokkadam]

According to the given function,
h(100) = 2*4*6*8*...*100

By factoring a 2 from each term of our function, h(100) can be rewritten as
2^50*(1*2*3*...*50).

Thus, all integers up to 50 - including all prime numbers up to 50 - are factors of h(100).

Therefore, h(100) + 1 cannot have any prime factors 50 or below, since dividing this value by any of these prime numbers will yield a remainder of 1.

Since the smallest prime number that can be a factor of h(100) + 1 has to be greater than 50, The correct answer is E. According to the given function,
h(100) = 2*4*6*8*...*100

By factoring a 2 from each term of our function, h(100) can be rewritten as
2^50*(1*2*3*...*50).

Thus, all integers up to 50 - including all prime numbers up to 50 - are factors of h(100).

Therefore, h(100) + 1 cannot have any prime factors 50 or below, since dividing this value by any of these prime numbers will yield a remainder of 1.

Since the smallest prime number that can be a factor of h(100) + 1 has to be greater than 50, The correct answer is E.

1) For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n inclusive. If p is the smallest prime factor of h(100) + 1, then p is between
a. 2 and 10
b. 10 and 20
c. 20 and 30
d. 30 and 40
e. > 40


2) Positive integers x, y, and z are such that x is a factor of y and y is a factor of z, if z even? (EXTRA DIFFICULT)
a. xz is even
b. y is even

[sidvenki]

2nd is a lot easier than the 1st one.

x is factor of y => y = k1*x
y is a factor of z => z = k2*y

1) Z is even if y is even. so statement B is sufficient to answer z is even.

2) Statement A is xz = even;

z = k2y = k1k2x;

Given as xz is even; either x is even or z is even or both.

Assuming x is even-> As z = k1k2x; Therefore x is even.

If x is odd then z is even from Statement A.

So Answer is D (Both Stmts can independently answer the question)

Sid

1) For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n inclusive. If p is the smallest prime factor of h(100) + 1, then p is between
a. 2 and 10
b. 10 and 20
c. 20 and 30
d. 30 and 40
e. > 40


2) Positive integers x, y, and z are such that x is a factor of y and y is a factor of z, if z even? (EXTRA DIFFICULT)
a. xz is even
b. y is even